Polyphase Filter Design with Reduced Phase Non-Linearity

Transcription

1 Polyphase Filter Design with Redued Phase on-linearity DR. ARTUR RUOWSI and DR. IZZET ALE Applied DSP and VLSI Researh Group, University of Westminster, 5 ew Cavendish Street, Londo UITED IGDOM Abstrat: - The perfet linear phase requirement an be ahieved only by diret design with an FIR filter at the expense of high filter order and hene high omputational burden for the given speifiation. Alternatively, IIR filters an be designed with muh smaller orders than their FIR ounterparts, but at the expense of the nonlinear phase. However, it is possible to design almost linear-phase IIR filters for a given speifiation of the allowed phase or group delay ripples. This paper presents an improvement to the existing algorithms for the design of almost linear phase, arbitrary-band IIR filters and gives a different viewpoint on these methods, showing a onsiderable improvement in maximum ripples of the magnitude and group delay responses. The idea of dereasing the phase non-linearity of the -path polyphase IIR filters with a ombination of allpass setions of different order and oeffiient polarity is presented. The effet of oeffiient quantizatio in view of the fixed-point implementatio on the performane of the resulting filter is also shown. ey-words: - Filter desig Phase-linearity, Digital filter, Polyphase struture, Multi-rate system. Introdution For systems, whih ater for wide-band signals, it is important to ensure the same time differenes between signal spetrum omponents before and after filtering. This requirement is met if the phase response of the filter is linear and has no onstant fator, or when the group delay funtion is onstant: ( ) ν ( ) φ ν π ν onst () A flatness of the group delay funtion is enough to assure the phase response linearity for the polyphase filter []-[3], shown in Fig., due to its monotoni phase response rossing the zero at DC. X(z) z - A (z ) A (z ) z -.. A - (z ) Σ Y(z) H ( z) A ( z) A n n n A A ( z) z + z ( z) + z n ( z) Fig. The general -path polyphase struture. The group delay of the struture is given by: + + πν κ κ os 4 κ ( ν ) + + δ ( ν z ) n () (3) The shape of the overall group delay funtion is an average of group delays of all allpass subfilters. The seond term in (3) is responsible for the phase jumps due to zeros of the filter on the init irle. The example shapes of the group delay for different oeffiients and setion orders are shown in Fig.. (ν).5 > < 4 > 4 <.5.5 ν Fig. Example group delay of the polyphase filter. The phase non-linearity an be dereased in two ways, either by designing the orretor with the phase response opposite to the filter s one or by designing the filter diretly for almost flat group delay. It an be notied from Fig. that for negative oeffiients the group delay is symmetri against the Y-axis to the one for the positive oeffiient. Therefore the orretion an be ahieved with a ombination of allpass filters having negative oeffiients. A asade of different order allpass setions an be used to ahieve a better orretion. The seond way of designing approximately linear-phase IIR filters is to modify the struture from Fig. by replaing the allpass filter in one of the branhes with a bul delayor, z -, ( being the

2 order of the allpass filter) as in Figure 3. The allpass filter A(z) represents the transfer funtion of allpass filters in other branhes of the struture. x(z) A(z) z - Σ +/-.5 y(z) Fig. 3 The struture of an approximately linearphase polyphase two-path filter. In order to ahieve lowpass filter response the allpass filter A(z) has to be designed in-phase with the bul delay at the low frequenies and distant by π at the frequenies lose to yquist (or other way round for the highpass filter). This way A(z) and hene the overall filter have an almost linear-phase. The idea as suh is not new and was suggested and used by Curtis [][4] and employed in designs published by Lawson [5] and Lu [6]. The urrent design methods are based on the standard idea of omposing two idential IIR (non-linear phase) filters to ahieve an approximately linear-phase harateristi [7] or applying iterative quadrature programming methods [6]. Suh an approah does not allow muh flexibility, limiting the number of points of freedom to half of what would be available when standard IIR filters are used. Thus the resulting filters posses larger stopband ripples as well as larger group delay ripples than what the struture is really apable of ahieving. The additional advantage of the presented struture is that it is apable of providing a omplimentary set of lowpass and highpass filters, simply by replaing the adder with a subtrator []-[3]. This feature is very attrative for applying suh filters in almost linearphase IIR quadrature filter bans. Phase Compensation with Casade of Allpass Setions Correting the group delay (phase linearity) of the polyphase filters with allpass setions A (z) from () having negative oeffiients is espeially appliable within frequeny bands, whih are the power of two divisions of the yquist frequeny, i.e..5,.5 et. As it was shown before in Fig., the shape of the allpass phase response having a negative oeffiient is opposite to the one having the positive one. As the total group delay of the polyphase struture is equal to the average of its allpass omponents, it has the same bell-lie shape, peaing at half-yquist for seond-order allpasses, quarter- yquist for forth-order ones, et. Here the ase of two-path struture () is disussed. For strutures with more paths the ompensation is very similar. The simplest orretor is a single seond-order allpass setion. The best orretion is ahieved when the group delay of the final filter at DC is equal to the group delay at its utoff frequeny, ν, i.e.: ( ν ) () ( ) o( ν ) o() + os4πν + (4) where are orretor oeffiients. The effet of suh orretion an be seen in Fig. 4 for a twooeffiient (.5,.55) two-path () filter. This single-oeffiient orretor give a 6.5 times derease of group delay pea-to-pea error in the signal band up to ν (ν) () Original filter ν.5 Resulting filter Allpass orretor Fig. 4 The group delay orretion with a single allpass setion for the example twooeffiient filter up to ν.5. The orretion an be improved by asading additional higher order allpass setions. In general arbitrary orders an be hosen. However, the best results are ahieved by for doubling of the onseutive orders of the allpass setions. Apart from symmetry of ompensatio this gives an additional advantage for multirate systems [8], [9]. The order of the seond ompensator setion should be hosen to be.5/ν, i.e. the overall -setion ompensator transfer funtion beomes: H + z ( ) ν + z m + z + z m ν m ν (5) Higher order ompensators an be designed by minimizing the sum of the pea differenes of the overall group delay squared with respet to an average value, as in Fig. 5, i.e.: C ( ν ), ( ν ) med med (6) m m ν

3 ν (ν) ν ν 3 Fig. 5 The non-linear optimisation ost funtion. It was notied for the number of orretor setions greater than two that the absolute values of oeffiients for onseutive setions approximately follow a geometri series. Therefore finding the first two setion oeffiients allows estimating a good starting point for the multi-setion orretor optimisation. The alulation of the seond setion oeffiient is explained in Fig (ν) () Final result Result of first ompensator, T(ν).65 ν 4 Seond allpass ompensator ν 5 Original filter Fig. 6 The orretion with two allpass setions. The oeffiient is alulated to satisfy (7): ν T T T ( ) +, ( ) ( ν ) ( ν ) + (, ν ) (, ν ) T min ν ν min min ν 6 ν 7 ν.5 ν (7) where T is the group delay of the filter ompensated with a single allpass setion. The the initial set of ompensator oeffiients for an optimisation is: init,,,, (8) The group delay orretion up to ν.5 for the same example two-oeffiient filter with four allpass setions is shown in Fig. 8. (ν) (ν) Original filter (ν)- (ν) Correted filter (ν)- (ν) Corretor (ν)- (ν).5 ν Fig. 8 Four oeffiient group delay orretion for the example polyphase LPF up to ν ut.5. The group delay pea-to-pea differene was dereased from.43 to.3e-4 - over three thousand times! The orretion results with the number of setions between one are four are shown in Table for both floating-point (FP) and fixed-point (FX) orretor oeffiients. Setions 3 4 FP bit FX bit FX bit FX Table Derease of the pea-to-pea group delay ripples using different number of allpasses. The fator is a ratio between the group delay pea-to-pea error before and after the orretion: ( ν ) max ( ν ) ( ν ) ( ν ) max min min ν < ν (9) As polyphase strutures are often used in high speed filtering appliation and most of the times are implemented in fixed-point arithmeti, it is required for the orretor oeffiients to be onstraint to short bit lengths. The onstraint orretor oeffiients were alulated from the floating-point ones by trunating them to four, eight and sixteen bits and reoptimising. He results are shown in Table. The performane of the orretion is very muh dependant on the bit length of the oeffiients. It dereases quily when shortening the wordlength of the oeffiients beause their values in onseutive allpass setions follow a geometri series, quily onverging to zero. For small bit lengths some of the oeffiients are too small for the given number of bits and the number of orretor oeffiients dereases. If the orretion performane beomes unsatisfatory, an alternative has to be sought. 3 Almost Linear-Phase IIR Filter Design Algorithm The alternative way to ahieve almost linear-phase IIR filter is to use the struture from Fig. 3. The implemented design routine was based on the disrete filter least squares fit to the frequeny response data ( invfreqz ) fro Matlab to approximate the phase of the filter in the passband to be z - and z --.5 in the filter stopband. The important fator was the hoie of the weighting funtion. Choosing onstant weights for the passband and stopband led to stopband ripples dereasing monotonially with frequeny with passband ripples monotonially

4 inreasing. Therefore an iterative update of weights was done at every iteration step of the optimisatio whih was hanging them aording to the shape of the envelope of the magnitude response and group delay ripples in the passband and in the stopband. In general ase the allpass filter used in the top branh of the struture from Fig. 3, does not neessarily have to be symmetri against ν.5. Also in order to ensure that both small passband ripples and high stopband attenuation is ahieved, it is important to monitor the group delay ripples both in the filter passband and its stopband. The design algorithm an be desribed as follows:. Speify the required omplex magnitude response shape to equal the response of the z - delayor within the filter passband and equal the response of a z --.5 delayor in the filter stopband. Also speify the frequeny grid in a logarithmi sale to be denser lose to the transition band.. Choose the initial weights, W(ν), for the optimization equal to unity at all frequenies. 3. Perform weighted least-squares fit to the frequeny response data. 4. Calulate overall group delay of the filter, (ν). 5. Calulate the maxima of the modulus of the group delay funtio (ν), and interpolate the new funtio (ν), approximating (ν) through these points. 6. Update the weights using: [ ] * ( ν ) [ + W ( ν )] + ( ν ) W () 7. ormalize the maximum value of the weights to unity and sale the rest of them aordingly. 8. If the iteration number is less than the maximum, proeed to point three, otherwise deliver the answer vetor. 3. Experimental Results During the experiments, it was found that a maximum of four iterations was required to ahieve the final result within % differene with regard to the result obtainable if iterations were to ontinue for infinite number of iterations. In order to measure the performane of the method it was ompared to the similar approahes suggested by Lu [6] and Lawson [5]. The example filters were designed aording to the speifiations given in these publiations. Comparative results showing the stopband attenuatio the magnitude response ripples and the group delay deviations for the given passband and stopband ut-off frequenies, ν p and ν s respetively, are given in Table. Design ν p, ν s [-] Lu 5..8 WLSq Lawson WLSq ε p [db] A [db] ( max - min )/ [samples] % / /.5 Table Comparison of the weighted least squared approah to approximating the linear phase of the polyphase LPF to the Lu s [6] one and the Lawson s [5] one. Plots of the filters designed through our method are shown in Fig. 9 and Fig.. It an be learly seen from these plots that the suggested weighted least square method of is advantageous when ompared both to [5] and [6] Top branh phase: LPF magnitude: Group Delay error: Im(z) Top Branh PZP Fig. 9 Example filter designed to the same speifiation as Lawson s one [5]. In both ases the filter order was hosen to be one less than the ones in the ompetitive designs and even so the suggested approah delivered muh better magnitude and group delay ripples both in the passband and in the stopband. There are two values given for the group delay ripples in Table. The first one is the maximum ripple value to within 96% of the bandwidth and the seond one for the full bandwidth. It an be learly seen from Fig. 9 and Fig that the stopband deviations are not equiripple. The purpose was to mae a ompromise between ahieving maximum attenuation and minimum group delay ripples in the passband. Re(z)

5 However, it was not possible to ahieve both equiripple group delay and equiripple maximum stopband attenuation [7]. It was notied that inreasing the requirements for the group delay ripples led to degradation in the stopband performane and vie versa. Dereasing the group delay ripples by a few perent was ausing degradation in the stopband attenuation by a few db. The suggested design method was tested on a number of examples for different ut-off frequenies and transition bandwidths. The best performane for the given filter order and transition band speifiation were ahievable for the ase of the halfband filter. In suh a ase the design too advantage of the symmetri allpass filter response, whih was easier to ahieve Top branh phase: LPF magnitude: Group Delay error: Top Branh PZP Im(z) Fig. Example filter designed to the same speifiation as Lu s one []. Design ew Lu ew Lawson [-] ν p, ν s [-] Re(z) ε p Α ( max - min) / [µdb] [db] [samples] / /.5 Table 3 Performane of the example filters with original Lu s and Lawson s speifiations with the utoff frequeny set at ν.5. In order to demonstrate this idea the filter speifiation from the previous example has been modified to save its utoff frequeny at ν.5. The performane of these example filters, whih were fored to be symmetri, is presented in Table 3. It an be seen that the stopband attenuation does not inrease muh when the filter is symmetri against ν.5 improving only by.5db for Lu s modified speifiation and not at all for Lawson s modified filter. This is beause the utoff frequeny in the original speifiation was very lose to ν.5. However, the phase response is muh more dependent on the symmetry of the transfer funtion of an allpass, whih aused a signifiant differene in the group delay ripples. The ripples of the group delay dereased by the fator of two for both filters. 3. Constraint oeffiient ase It has been shown in a number of publiations [8], [9] that the polyphase IIR struture as given in []-[3] is very attrative for onstraining of its oeffiients. This is true also for the speial ase of the polyphase IIR struture as presented in this paper. Additionally, for the ase of the half-band filter the allpass in the top branh of Figure an be easily deomposed into the asade of seond-order oneoeffiient allpass setions. Then for the implementation of suh a filter, the required number of multipliations would be equal to half of the filter order. The oeffiients of these filters an be subsequently onstraint to a limited wordlength and re-optimised for the best performane using, for example, the bit-flipping algorithm []. The effet of onstraining the oeffiients of the filter having Lu s speifiation [6] to 4, 8 and bits is presented in Figure H(ν) db 4 bits bits 8 bits ν Fig. The magnitude response of the filter designed to Lu s speifiation [6] with limited oeffiient wordlengths. umber of bits ε p [µdb] Α [db] ( max - min) / [samples] Table 4 Performane of the filter designed to Lu s speifiation with onstrained oeffiients.

6 The filter attenuatio magnitude and group delay ripples in the passband are summarized in Table 3. For these tests all the oeffiients were trunated to the required wordlength without re-optimization. Even so, the results were very lose to the floatingpoint version. It should be noted here that these are not oeffiients of the transfer funtio but of the seond and fourth-order setions into whih the filter has been deomposed. Constraining the oeffiients of the filter with arbitrary ut-off frequeny maes the problem similar to onstraining the taps of the general transfer funtion of an IIR. Hene the performane is muh more sensitive to shortening of the oeffiient wordlength. The simulation result showed that even for short oeffiient wordlength, suh as 6-bits, the responses were very lose to the floating-point one. These results further suggest that for the halfband ase implementing the allpass filter from Figure as the asade of smaller order allpass setions maes the oeffiients muh less sensitive to limiting their wordlengths. 4 Conlusion We have presented in this paper two methods of designing lowpass/highpass filters based on the polyphase IIR struture ahieving almost linear phase response. One was to ompensate the non-linear phase of the standard polyphase filter with a ombination of one-oeffiient allpass filters with different order. This type of orretion of the phase linearity allows ahieving very high improvement ratios. For the ase of frational passbands the even orders of the allpass orretion setions mae them suitable for operating at a lower rate in multi-rate systems. The seond approah was to use Curtis [][4] flat group delay struture. An allpass filter in one branh was designed to follow the linear phase response of the bul delay plaed in the other branh. It has been ahieved with weighted least squares algorithm with ustom weight optimisation. This approah was ompared to similar wor by Lu and Lawso and proved advantageous to them in terms of the required filter order, stopband attenuation and ahieved group delay flatness. It was also shown that the implemented struture allows very effiient oeffiient quantizatio espeially for the ase of the halfband filter. This maes it very appliable for fixed-point implementation in very high speed filtering appliations lie Σ -deimation filters and other appliations, whih neessitate a high level of phase or group delay flatness. Referenes: [] Renfors, M. and T. Saramäi, "Reursive thband digital Filters - Part II: Design of multistage deimators and interpolators," IEEE Transations on Ciruits and Systems, vol. CAS-34, pp. 4-5, January 987. [] harris, f., M. d'oreye de Lantremange and A. G. Constantinides, "Digital signal proessing with effiient polyphase reursive all-pass filters", International Conferene on Signal Proessing, Florene, 4-6 September 99. [3] Valenzuela, R. A. and A. G. Constantinides, "Digital signal proessing shemes for effiient interpolation and deimation", IEE Proeedings, Pt. G., vol. 3, no. 6, pp. 5-35, Deember 983. [4] Curtis, T. E. and A. B. Webb, "High performane signal aquisition systems for sonar appliations", International Conferene on A/D and D/A Conversio Swansea, 7-9 September 99. [5] Lawso S., On design tehniques for approximately linear phase reursive digital filters, Proeedings International Conferene on Ciruits and Systems (ISCAS 97), pp. - 5, June 997. [6] Lu, W. S., Design of stable IIR digital filters with equiripple passbands and pea-onstrained least squares stopbands, Proeedings International Conferene on Ciruits and Systems (ISCAS 97), pp. 9-95, June 997. [7] Lawso S. S., Diret approah to design of PCAS filters with ombined gain and phase speifiation, IEEE Proeedings on Visual Image Signal Proessing, vol. 4, pp. 6-67, 994. [8] ale, I., R. C. S. Morling and A. ruowsi, "A high-fidelity deimator hip for the measurement of Sigma-Delta modulator performane", IEEE Transations on Instrumentation and Measurement, vol. 44, no. 5, Otober 995. [9] ruowsi, A. and I. ale, "Constrained oeffiient variable ut-off polyphase deimation filters for band-pass Sigma-Delta data onversion IMEO Worshop on ADC Modeling, House of Sientists, Smolenie Castle, 7-9 May 996. [] ruowsi, A., I. ale,. Hejn and G. D. Cai "A bit-flipping approah to multistage two-path deimation filter design", Seond International Symposium on DSP for Communiation Systems, SPRI, 6-9 April 994.